Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences of $f_1$ and $f_2$.

For instance, suppose that we have maps $g_X : C^*(X_1) \to C^*(X_2)$ and $g_Y: C^*(Y_1) \to C^*(Y_2)$ of singular cochain complexes (not necessarily coming from topological maps), such that the following diagram commutes: $\require{AMScd}$ \begin{CD} C^*(Y_1) @>f_1^*>> C^*(X_1)\\ @Vg_YVV @VVg_XV\\ C^*(Y_2) @>f_2^*>> C^*(X_2). \end{CD}

Does this give rise to map between the Leray spectral sequences of $f_1$ and $f_2$ ?

More generally, I'm looking for a reference where the construction of the Leray spectral sequence is done entirely. So far I have seen two constructions: one rather categorical with Grothendieck spectral sequence, and the other one using the Cech-DeRham double complex (Bott-Tu). Both constructions are problematic for me because I'm dealing with quite general topological spaces (so Cech-DeRham is not relevant), and I work with maps at the level of cochain complexes, so Grothendieck spectral sequence doesn't really help.

Thanks a lot.